Determining all 6 trig ratios in a triangle

Trigonometric ratios describe the relationships between the different sides of a right-angled triangle. These sides are described in terms of where they are compared to one of the two acute angles.

So a trigonometric ratio tells us which two sides are in a given ratio, as well as the angle the ratio is being measured from.

In the triangle below we’ve chosen one of the acute angles, labelled $x$x. The sides of the triangle can then be referred to as the side opposite $x$x, the side adjacent to $x$x and the hypotenuse.

Some trigonometric ratios of $x$x are as follows:

$\sin\left(x\right)=\frac{\text{Opposite}}{\text{Hypotenuse}}$sin(x)=OppositeHypotenuse​

$\cos\left(x\right)=\frac{\text{Adjacent}}{\text{Hypotenuse}}$cos(x)=AdjacentHypotenuse​

$\tan\left(x\right)=\frac{\text{Opposite}}{\text{Adjacent}}$tan(x)=OppositeAdjacent​

There are also what we call the reciprocal trigonometric ratios, each of which are a reciprocal of one of the three ratios above.

$\csc\left(x\right)=\frac{1}{\sin\left(x\right)}=\frac{\text{Hypotenuse}}{\text{Opposite}}$csc(x)=1sin(x)​=HypotenuseOpposite​

$\sec\left(x\right)=\frac{1}{\cos\left(x\right)}=\frac{\text{Hypotenuse}}{\text{Adjacent}}$sec(x)=1cos(x)​=HypotenuseAdjacent​

$\cot\left(x\right)=\frac{1}{\tan\left(x\right)}=\frac{\text{Adjacent}}{\text{Opposite}}$cot(x)=1tan(x)​=AdjacentOpposite​

Notice that these ratios describe a relationship between the sides of the triangle rather than the exact length of each side. Enlarging or shrinking a triangle won't change the trigonometric ratio of a particular angle, because enlarging or shrinking a triangle doesn't change the size of its angles or the ratio of its sides. 
For example, if $\tan x=\frac{2}{1}$tanx=21​ we can tell that the side opposite to $x$x is twice as big as the side adjacent to $x$x. The sides might have lengths $6$6 and $3$3, not necessarily $2$2 and $1$1.

Even though we don’t know the exact length of the sides just from knowing these ratios, we can still use one ratio to calculate another ratio.

Practice Questions

Question 1

Question 2

Question 3